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Applied Mathematics Letters 19 (2006) 13611366

www.elsevier.com/locate/aml

Periodic solutions for discrete predatorprey systems with theBeddingtonDeAngelis functional response

Jimin Zhang, Jing Wang

School of Mathematics and Statistics, KLAS and KLVE, Northeast Normal University, 5268 Renmin Street, Changchun, Jilin, 130024, PR China

Received 18 April 2005; received in revised form 22 January 2006; accepted 15 February 2006

Abstract

Sufficient criteria are established for the existence of positive periodic solutions of discrete nonautonomous predatorprey

systems with the BeddingtonDeAngelis functional response using a continuation theorem.

c 2006 Elsevier Ltd. All rights reserved.

Keywords: Periodic solutions; Difference equations; Predatorprey; BeddingtonDeAngelis functional response; Coincidence degree

1. Introduction

Predatorprey interaction is one of the major forces shaping food webs. Since the great work of Lotka (in 1925)

and Volterra (in 1926), modelling these interactions has been one of the central themes in mathematical ecology.

One significant component of the predatorprey relationship is the predators rate of feeding upon prey, i.e., the so-called predators functional response. In general, the functional responses can be either prey dependent or predator

dependent. Functional response equations that are strictly prey dependent, such as the Holling family ones, arepredominant in the literature.

However, the prey-dependent ones fail to model the interference among predators, and have been facing challenges

from the biology and physiology communities. The predator-dependent functional responses can provide better

descriptions of predator feeding over a range of predatorprey abundances, as is supported by much significantlaboratory and field evidence (see [5] and references therein). The BeddingtonDeAngelis functional response,

first proposed by Beddington [3] and DeAngelis [2], performed even better. So, predatorprey systems with theBeddingtonDeAngelis response have been studied extensively in the literature [1,6,7,9,10].

It is well known that the discrete time models governed by difference equations are more appropriate than the

continuous ones when the populations have nonoverlapping generations. In addition, discrete time models can also

provide efficient computational models of continuous for numerical simulations. However, no such work has beendone for predatorprey systems with the BeddingtonDeAngelis functional response.

Corresponding author.E-mail address: Zhangjm588@nenu.edu.cn (J. Zhang).

0893-9659/$ - see front matter c 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.aml.2006.02.004

http://www.elsevier.com/locate/amlmailto:Zhangjm588@nenu.edu.cnhttp://dx.doi.org/10.1016/j.aml.2006.02.004http://dx.doi.org/10.1016/j.aml.2006.02.004mailto:Zhangjm588@nenu.edu.cnhttp://www.elsevier.com/locate/aml

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1362 J. Zhang, J. Wang / Applied Mathematics Letters 19 (2006) 13611366

In this work, we will study the existence of positive periodic solutions of the discrete analogue of the predatorprey

system with BeddingtonDeAngelis functional response explored by Fan and Kuang [ 5]. Following the clues in [4],

with the help of differential equations with piecewise constant arguments, one can reach its discrete analogue

x (k+ 1) = x (k) exp

a(k) b(k)x (k)

c(k)y(k)

(k) + (k)x (k) + (k)y(k)y(k + 1) = y(k) exp

d(k) +

f(k)y(k)

(k) + (k)x(k) + (k)y(k)

, k Z (1.1)

where Z denotes the set of integers. In the following, we will focus our attention on system (1.1). Considering the

biological significance, we consider (1.1) with positive initial values and assume that the parameters in system (1.1)

are nonnegative and -periodic with Z and > 1.

2. Existence of positive periodic solutions

Let Z+, R+ and R2 denote the set of nonnegative integers, nonnegative real numbers, and two-dimensional

Euclidean vector space, respectively, and let

Iw = {0, 1, 2, . . . , w 1} g = 1w

w1k=0

g(k) gu = maxkIw

g(k) gl = minkIw

g(k)

where g(k) is a w-periodic sequence of nonnegative real numbers defined for k Z.Let X, Y be normed vector spaces, L : Dom L X Y be a linear mapping, N : X Y be a continuous

mapping. The mapping L will be called a Fredholm mapping of index zero if dim Ker L = codim Im L < + and

Im L is closed in Y. If L is a Fredholm mapping of index zero and there exist continuous projections P : X X andQ : Y Y such that Im P = Ker L , Im L = Ker Q = Im(I Q), it follows that L|Dom L Ker P : (I P)X

Im L is invertible. We denote the inverse of that map by KP . If is an open bounded subset ofX, the mapping N will

be called L-compact on ifQN() is bounded and KP (I Q)N : X is compact. Since Im Q is isomorphic

to Ker L, there exists an isomorphism J : Im Q Ker L.

Lemma 2.1 (Continuation Theorem [8]). Let L be a Fredholm mapping of index zero and N be L-compact on .Suppose:

(a) for each (0, 1), every solution x of Lx = Nx is such that x ;(b) QNx = 0 for each x Ker L and the Brouwer degree deg{JQN, Ker L , 0} = 0.

Then the operator equation Lx = Nx has at least one solution lying in Dom L .

Lemma 2.2 ([4, Lemma 3.2]). Let g : Z R be w periodic, i.e., g(k + w) = g(k). Then for any fixed k1, k2 Iw,

and any k Z, one has

g(k) g(k1) +

w1

s=0

|g(s + 1) g(s)| g(k) g(k2) w1

s=0

|g(s + 1) g(s)|.

Define l2 = {u = u(k) : u(k) R2, k Z+}. Let lw l2 denote the subspace of all w periodic sequences with

the usual supremum norm , i.e.,

u = maxkIw

|u1(k)| + maxkIw

|u2(k)|, for any u = {u(k) : k Z+} lw.

It is not difficult to show lw is a finite-dimensional Banach space. Let

lw0 =

u = {u(k)} lw :

w1k=0

u(k) = 0, k Z+

lwc = {u = {u(k)} l

w : u(k) = h R2, k Z+};

then it follows that lw0 and lwc are both closed linear subspaces ofl

w and lw = lw0 lwc , dim lwc = 2, so we reach ourmain result as follows:

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J. Zhang, J. Wang / Applied Mathematics Letters 19 (2006) 13611366 1363

Theorem 2.1. If a > (c/ ) and( f du )(a (c/))b1 exp{2a} du > 0, then system (1.1) has at least one

positive periodic solution.

Proof. First, let x(k) = exp{u1(k)}, y(k) = exp{u2(k)}, so that (1.1) becomes

u1(k + 1) u1(k) = a(k) b(k) exp{u1(k)} c(k) exp{u2(k)}

(k) + (k) exp{u1(k)} + (k) exp{u2(k)}

u2(k + 1) u2(k) = d(k) +f(k) exp{u1(k)}

(k) + (k) exp{u1(k)} + (k) exp{u2(k)}.

(2.1)

Define X = Y = lw, (Lu )(k) = u(k + 1) u(k), and

N u = N

u1u2

(k) =

a(k) b(k) exp{u1(k)} c(k) exp{u2(k)}

(k) + (k) exp{u1(k)} + (k) exp{u2(k)}

d(k) +f(k) exp{u1(k)}

(k) + (k) exp{u1(k)} + (k) exp{u2(k)}

for any u X and k Z+. It is trivially easy to see that L is a bounded linear operator and Ker L = lwc , Im L = lw0 ,

as well as that dim Ker L = 2 = codim Im L. Since Im L is closed in Y, it follows that L is a Fredholm mapping ofindex zero. Define

Pu =1

w

w1s=0

u(s), u X, Qz =1

w

w1s=0

z(s), z Y.

It is easy to show that P and Q are continuous projections such that Im P = Ker L and Im L = Ker Q = Im(I Q).Then, the generalized inverse (to L) KP : Im L Ker P

Dom L exists and is given by

KP (z) =

w1s=0

z(s) 1

w

w1s=0

(w s)z(s).

Obviously, Q N and KP

(I Q)N are continuous. Since X is a finite-dimensional Banach space, on the basis of the

ArzelaAscoli theorem, it is not difficult to show that KP (I Q)N() is compact for any open bounded set X.

Furthermore, Q N() is bounded, so N is L-compact on with any open bounded set X.

For the application of the continuation theorem, we must search for an appropriate open, bounded subset .

Corresponding to the operator equation Lu = N u, (0, 1), we have

u1(k + 1) u1(k) =

a(k) b(k) exp{u1(k)}

c(k) exp{u2(k)}

(k) + (k) exp{u1(k)} + (k) exp{u2(k)}

u2(k + 1) u2(k) =

d(k) +

f(k) exp{u1(k)}

(k) + (k) exp{u1(k)} + (k) exp{u2(k)}

.

(2.2)

Assume that u(k) = {(u1(k), u2(k))T} X is an arbitrary solution of system (2.2) from 0 to w 1 with respect to k;

we reach

aw =

w1k=0

b(k) exp{u1(k)} +

c(k) exp{u2(k)}

(k) + (k) exp{u1(k)} + (k) exp{u2(k)}

dw =

w1k=0

f(k) exp{u1(k)}

(k) + (k) exp{u1(k)} + (k) exp{u2(k)}

(2.3)

From (2.2) and (2.3), we obtain

w1

k=0|u1(k + 1) u1(k)|

w1

k=0

a(k) + b(k) exp{u1(k)} +c(k) exp{u2(k)}

(k) + (k) exp{u1(k)} + (k) exp{u2(k)}

= 2aw

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1364 J. Zhang, J. Wang / Applied Mathematics Letters 19 (2006) 13611366

w1k=0

|u2(k+ 1) u2(k)| w1k=0

d(k) +

f(k) exp{u1(k)}

(k) + (k) exp{u1(k)} + (k) exp{u2(k)}

= 2 dw.

Since (u1(k), u2(k)) X, there exist i , i Iw , such that

ui (i ) = minkIw

{ui (k)} ui (i ) = maxkIw

{ui (k)}, i = 1, 2. (2.4)

It follows from (2.3) and (2.4) that

aw

w1k=0

b(k) exp{u1(k)} w1k=0

b(k) exp{u1(1)} = exp{u1(1)}bw

which reduces to u1(1) ln(a/b) := L1, and hence

u1(k) u1(1) +

w1

s=0|u1(s + 1) u1(s)| ln{a/b} + 2aw := H1. (2.5)

On the other hand, from (2.3) and (2.4), we also have

aw

w1k=0

b(k) exp{u1(1)} +

c(k)

(k)

= (c/)w + bw exp{u1(1)}

which reduces to u1(1) ln{a(c/ )

b} := l1, and hence

u1(k) u1(1)

w1s=0

|u1(s + 1) u1(s)| ln

a (c/ )

b

2aw := H2

which, together with (2.5), leads to maxkIw |u(k)| max{|H1|, |H2|} := B1.From (2.3) and (2.4), we obtain

dw

w1k=0

f(k) exp{u1(k)}

l exp{u1(k)} + l exp{u2(k)}

w1k=0

f(k)(a/b) exp{2aw}

l (a/b) exp{2aw} + l exp{u2(2)}.

Then u2(2) ln{( fdl )a exp{2aw}

b dl} := L2; we have

u2(k) u2(2) +

w1s=0

|u2(s + 1) u2(s)| ln

( f dl )a exp{2aw}

b dl

+ 2 dw := H3. (2.6)

We also can obtain from (2.3) that

dw

w1k=0

f(k) exp{u1(k)}

u + u exp{u1(k)} + u exp{u2(2)}

w1k=0

f(k)(a (c/))b1 exp{2aw}

u + ua(c/ )

bexp{2aw} + u exp{u2(2)}

.

Then u2(2) ln{( fdu )(ac/ )b1 exp{2aw}du

du} := l2; it follows that

u2(k) u2(2)

w1s=0

|u2(s + 1) u2(s)| ln

( f du )(a c/ )b1 exp{2aw} du

du

2 dw := H4

so maxkIw |u2(k)| max{|H3|, |H4|} = B2. Obviously, B1 and B2 are independent of. Take B = B1 + B2 + B0

where B0 is taken sufficiently large that B0 > |l1| + |L1| + |l2| + |L2|.

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J. Zhang, J. Wang / Applied Mathematics Letters 19 (2006) 13611366 1365

Fig. 1. A solution (1.1) with a(k) = 1 + 0.2 sin( k2

), b(k) = 0.002(2 + sin( k2

)), c(k) = 0.5 + 0.2 cos( k2

), d(k) = 0.6 + 0.1 sin( k2

), f(k) =

0.8(0.5 + 0.2cos( k2

)),(k) = 0.2 + 0.1 cos( k2 ),(k) = 0.2 + 0.1 sin( k2

) , (k) = 1 and initial conditions x(0) = 20, y(0) = 5. The solution

tends to the 4-periodic solution (x(k), y(k)).

Considering the algebraic equations

a b exp{u1}

1

w1

k=0

vc(k) exp{u2}

(k) + (k) exp{u1} + (k) exp{u2}

d +1

w1k=0

f(k) exp{u1}

(k) + (k) exp{u1} + (k) exp{u2}

=

00

(2.7)

where v [0, 1] is a parameter and (u1, u2) R2. Similarly, one can easily show that any solution (u1, u

2) of(2.7)

with v [0, 1] satisfies

l1 u1 L1 l2 u

2 L2. (2.8)

Let = {(u1, u2)T X|(u1, u2) < B}; then is an open, bounded set in X and verifies requirement (a)

of Lemma 2.1. When (u1, u2)

Ker L =

R2, (u1, u2) is a constant vector in R

2 with (u1, u2) =

|u1| + |u2| = B . Then we have Q N[(u1, u2)T] = 0; that is, the first part of(b) of Lemma (2.1) is valid.

Consider the homotopy for computing the Brouwer degree

A((u1, u2)T) = Q N((u1, u2)

T) + (1 )F((u1, u2)T), [0, 1],

where

F((u1, u2)T) =

a b exp{u1}

d 1

w1k=0

f(k) exp{u1}

(k) + (k) exp{u1} + (k) exp{u2}

.

By the invariance property of homotopy, direct calculation produces

degJQN,Ker L, 0 = degQN,Ker L, 0 = degF, Ker L , 0 = 0,where deg(, , ) is the Brouwer degree and J is the identity mapping since Im Q = Ker L. We have proved that

verifies all requirements ofLemma 2.1; then Lx = Nx has at least one solution in Dom L

, so system (2.1) has at

least one periodic solution in Dom L

, say (u1(k), u2(k))

T. Let x (k) = exp{u1(k)}, y(k) = exp{u2(k)}, so

(x (k), y(k))T is an periodic solution of system (1.1) with strictly positive components. The proof is complete.

Remark 2.1. In (1.1), we have assumed that > 1. In fact, if = 1, then the parameters in (1.1) are all constants

and (1.1) is autonomous. The 1-periodic solutions of(1.1) must be constant solutions, i.e., the equilibria of (1.1).

Finally, in order to illustrate some features of our main results, let

a(k) = 1 + 0.2sink

2

, b(k) = 0.002

2 + sin k

2

, c(k) = 0.5 + 0.2cos k

2

,

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1366 J. Zhang, J. Wang / Applied Mathematics Letters 19 (2006) 13611366

d(k) = 0.6 + 0.1sin

k

2

, f(k) = 0.8

0.5 + 0.2cos

k

2

,

(k) = 0.2 + 0.1cos

k

2

, (k) = 0.2 + 0.1sin

k

2

, (k) = 1.

It is trivial to show that the conditions in Theorem 2.1 are verified. Therefore, (1.1) admits at least one -periodic

solution. Our numerical simulation supports our theoretical findings (see Fig. 1).

Acknowledgements

The authors would like express their gratitude to Prof. Meng Fan for helpful discussion and suggestions, and to

the referees for their excellent comments, which greatly improved the presentation of the work. The first author was

supported by the NSFC.

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[2] D.L. DeAngelis, R.A. Goldstein, R.V. ONeill, A model for trophic interaction, Ecology 56 (1975) 881892.[3] J.R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, J. Animal Ecol. 44 (1975) 331340.

[4] M. Fan, K. Wang, Periodic solutions of a discrete time nonautonomous ratio-dependent predatorprey system, Math. Comput. Modelling 35

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[5] M. Fan, Y. Kuang, Dynamics of a nonautonomous predatorprey system with the BeddingtonDeAngelis functional response, J. Math. Anal.

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[6] R.S. Cantrell, C. Cosner, On the dynamics of predatorprey models with the BeddingtonDeAngelis functional response, J. Math. Anal. Appl.

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[7] R.S. Cantrell, C. Cosner, Effects of domain size on the persistence of populations in a diffusive food chain model with DeAngelisBeddginton

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[8] R.E. Gaines, R.M. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Springer-Verlag, Berlin, 1977.

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